This demonstrates the Padua transform and inverse transform, explaining precisely the normalization and points
We define the Padua points and extract Cartesian components:
N = 15 pts = paduapoints(N) x = pts[:,1] y = pts[:,2]; nothing #hide
We take the Padua transform of the function:
f = (x,y) -> exp(x + cos(y)) f̌ = paduatransform(f.(x , y)); nothing #hide
and use the coefficients to create an approximation to the function $f$:
f̃ = (x,y) -> begin j = 1 ret = 0.0 for n in 0:N, k in 0:n ret += f̌[j]*cos((n-k)*acos(x)) * cos(k*acos(y)) j += 1 end ret end
#3 (generic function with 1 method)
At a particular point, is the function well-approximated?
f̃(0.1,0.2) ≈ f(0.1,0.2)
Does the inverse transform bring us back to the grid?
ipaduatransform(f̌) ≈ f̃.(x,y)
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